We can define a subclass of the regular languages. Fix an alphabet $\Sigma$. Define the "circular" languages (actually, the name already exists to denote a different thing it seems, used in the field of DNA computing. **AFAICT**, that's a different class of languages).

A language $L$ is circular if and only if for all words $w \in \Sigma$, we have:

$w\in L$ if and only if, for all integers $k > 0$ we have $w^k\in L$.

Is this class of languages known? I am interested in:

a name for it

decidability of the problem, given an automaton (in particular: a DFA), whether the accepted language obeys to the above definition

a "nice" characterization (e.g. equational?) of the definition.